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The sum product algorithm on factor graphs (FG/SPA) is a widely used tool to solve various problems in a wide area of fields. A representation of generally-shaped continuously valued messages in the FG/SPA is commonly solved by a proper parameterization of the messages. Obtaining such a proper parameterization is, however, a crucial problem in general. The paper introduces a systematic procedure for obtaining a scalar message representation with well-defined fidelity criterion in a general FG/SPA. The procedure utilizes a stochastic nature of the messages as they evolve during the FG/SPA processing. A Karhunen-Loève Transform (KLT) is used to find a generic canonical message representation which exploits the message stochastic behavior with mean square error (MSE) fidelity criterion. We demonstrate the procedure on a range of scenarios including mixture-messages (a digital modulation in phase parametric channel). The proposed systematic procedure achieves equal results as the Fourier parameterization developed especially for this particular class of scenarios.

Karhunen-Loève-Based Reduced-Complexity Representation of the Mixed-Density Messages in SPA on Factor Graph and Its Impact on BER